Table 1 Energy scales of the QMBS.

From: Two-dimensional hard-core Bose–Hubbard model with superconducting qubits

Energy scale

Description

\({\omega }_{{\rm{q}}}=\overline{{}_{i}\left\langle {\rm{e}}\right|\hat{H}{\left|{\rm{e}}\right\rangle }_{i}}\)

Qubit frequency

\(A=\overline{{}_{i}\left\langle {\rm{f}}\right|\hat{H}{\left|{\rm{f}}\right\rangle }_{i}}-2{\omega }_{{\rm{q}}}\)

Anharmonicity

\(J=\overline{{\left.\right|}_{i}{\left\langle {\rm{e}}| \hat{H}| {\rm{e}}\right\rangle }_{j}\left.\right|}\)

Hopping energy

\(\Delta \omega =\sqrt{\overline{{\left(_{i}{\left\langle {\rm{e}}| \hat{H}| {\rm{e}}\right\rangle }_{i}-{\omega }_{{\rm{q}}}\right)}^{2}}}\)

Frequency variance

  1. Here, \({\left|{\rm{e}}\right\rangle }_{i}\) (\({\left|{\rm{f}}\right\rangle }_{i}\)) is the state with qubit i in its first (second) excited state and all others in the ground state. \(\overline{{X}_{i}}\) (\(\overline{\overline{{X}_{i,j}}}\)) denote the average of Xi (Xi,j) over all qubits (all coupled pairs). We take ħ = 1 and the ground state energy to be zero.